Plotting the graph of the quadratic function and examining the roles of the parameters in the function of the form
Plotting the graph of the quadratic function and examining the roles of the parameters in the function of the form
– the coefficient of .
– the coefficient of .
– the constant term.
\( y=x^2 \)
Let's present the quadratic function and understand the meaning of each parameter in it.
– the coefficient of
- -determines if the parabola will be a maximum or minimum parabola (sad or smiling) - determines the steepness – its width.
must be different from 0.
If is positive – the parabola is a minimum parabola – smiling
If is negative – the parabola is a maximum parabola – sad
The larger is – the narrower the function will be and vice versa.
- the coefficient of
can be any number
indicates the position of the parabola along with .
determines the slope of the function at the intersection point with the axis
(not relevant to the material)
– the constant term
is not dependent on
can be any number
indicates the position of the parabola along with
determines the y-intercept
responsible for the vertical shift of the function
Wonderful. Now, let's move on to plotting the quadratic function.
Let's see an example -
In the following function:
\( y=x^2+10x \)
\( y=x^2-6x+4 \)
\( y=2x^2-5x+6 \)
Here we have a quadratic equation.
A quadratic equation is always constructed like this:
Where a, b, and c are generally already known to us, and the X and Y points need to be discovered.
Firstly, it seems that in this formula we do not have the C,
Therefore, we understand it is equal to 0.
a is the coefficient of X², here it does not have a coefficient, therefore
is the number that comes before the X that is not squared.
In fact, a quadratic equation is composed as follows:
y = ax²-bx-c
That is,
a is the coefficient of x², in this case 2.
b is the coefficient of x, in this case 5.
And c is the number without a variable at the end, in this case 6.
What is the value of the coefficient in the equation below?
The quadratic equation of the given problem is already arranged (that is, all the terms are found on one side and the 0 on the other side), thus we approach the given problem as follows;
In the problem, the question was asked: what is the value of the coefficientin the equation?
Let's remember the definitions of coefficients when solving a quadratic equation as well as the formula for the roots:
The rule says that the roots of an equation of the form
are :
That is the coefficientis the coefficient of the term in the first power -We then examine the equation of the given problem:
That is, the number that multiplies
is
Consequently we are able to identify b, which is the coefficient of the term in the first power, as the number,
Thus the correct answer is option d.
8
What is the value of the coefficient in the equation below?
The quadratic equation of the given problem has already been arranged (that is, all the terms are on one side and 0 is on the other side) thus we can approach the question as follows:
In the problem, the question was asked: what is the value of the coefficientin the equation?
Let's remember the definition of a coefficient when solving a quadratic equation as well as the formula for the roots:
The rule says that the roots of an equation of the form
are:
That is the coefficient
is the free term - and as such the coefficient of the term is raised to the power of zero -(Any number other than zero raised to the power of zero equals 1:
)
Next we examine the equation of the given problem:
Note that there is no free term in the equation, that is, the numerical value of the free term is 0, in fact the equation can be written as follows:
and therefore the value of the coefficient is 0.
Hence the correct answer is option c.
0